Bayes’ Theorem and Structural Reliability

Reliability-based method has been widely used by many researchers as well as design codes for evaluating the structural safety, in which the failure probability of structures is defined through the limit state functions. For the purpose of structural life-time management optimization, it is of critical importance to predict the structural reliability in the future based on the prior and present information that are obtained from various sources such as expert opinion, mathematical model, measurement data, existing laboratory and operational data, etc. Due to the large uncertainties that are inevitably associated with all those information, the probabilistic inference schemes are always more suitable for updating the time-dependent structural reliability. One such framework that enables the probabilistic inference in the context of reliability analysis is the Bayes’ theorem. In the past several decades, the Bayes’ theorem has become a widely used method due to its ability to learn and calibrate model by using the new information from the inspection, observations, etc.

1.2.4.1 Bayes’ Theorem

The Bayes’ theorem has been widely used for a wide range of applications in various engineering fields

such as structural monitoring, inspection, maintenance, and repair planning. Applications of Bayesian inference in civil engineering include structural reliability [101], structural identification [102], fatigue

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crack growth prediction [103], strength degradation of deteriorating concrete bridges [104], structural health monitoring and damage detection [105]. Enright and Frangopol [104] predicted the time-variant system reliability of a highway bridge and then evaluated the bridge condition by incorporating inspection results through Bayesian updating. Later on, Estes and Frangopol [106] applied the visual inspection data from the bridge management system to update the lifetime bridge reliability assessment. Zhang and Mahadevan [107] proposed a Bayesian procedure to quantify the modeling uncertainty using nondestructive inspections. Maes [108] established a stochastic deterioration model based on a discrete empirical Bayesian method that allows for a probabilistic reliability assessment of a reinforced concrete slab subject to long- term chloride corrosion by using the available inspection data. Beck and Katafygiotis [102,109] presented a Bayesian probabilistic system identification framework for structural model updating and the associated uncertainties quantification. Later on, as an extension of their work, Yuen et al. [105] applied a Bayesian time-domain approach for damage detection, location and assessment of a 15-story building using noisy incomplete excitation and response data. The proposed approach allows for the calculation of the probability of damage of different severity levels in each substructure, based on the updated probability density functions (PDFs) from data in the undamaged state and in a possibly damaged state. Recently, Straub and Papaioannou [101] proposed a method, termed as Bayesian updating with structural Reliability methods (BUS), to enable robust and efficient Bayesian updating of mechanical and other computational models. To summarize, through the application of Bayesian framework, the uncertain and incomplete information from either inspection data or engineering judgment can be combined and used with existing model in a rational manner, which enables better predictions for future structural conditions [104].

1.2.4.2 Bayesian Network

In real engineering applications, the reliability analysis in the component level as well as in the system level are required, as complex structures/systems with multiple components or multiple failure mechanisms are often involved. Although efforts have been made to update the system-level reliability using Bayes’ theorem, the earlier Bayesian updating methods for system reliability are usually unable to provide information on component performance [110,111]. The Bayesian network (BN) methodology is a well-

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suited framework for system reliability assessment, because it enables the uncertainty propagation and updating through nodes/components in the network that allows the transform of information from the system level to component level.

Figure 1.2 Example of BN

A BN, also called a Bayesian belief network (BBN), is a probabilistic model based on directed acyclic graph (DAG) that represents a joint probability distribution among a set of variables. As shown in Fig. 1.2, BNs are graphically mathematical models, where each node denotes a stochastic variable or a deterministic parameter of interest, and the links denote informational or causal dependencies among the nodes. Pearl [112] proposed the BNs in 1988 and the BNs have been originally developed and applied most in the field of artificial intelligence as an efficient and robust framework for reasoning with uncertain knowledge. BNs have many appealing features such as semantic clarity, the ease of acquisition and incorporation of prior knowledge, the possibility of causal interpretation of learned models and the automatic handling of noisy and missing data [113,114]. Later on, due to the possibility of system reliability updating with newly provided evidence, the BNs have become a popular modeling framework for system reliability analysis in the field of mechanical and civil structures in early 1990s. Almond [115] applied the GRAPHICAL- BELIEF tool, one of the early attempts of using BNs as framework for reliability analysis, to calculate the reliability of a low pressure coolant injection system for a nuclear reactor. Mahadevan et al. [110] proposed a BN model that incorporates multiple failure sequences and correlations between component failures for structure system reliability assessment and validated it with traditional reliability analysis. Langseth and Portinale [116] reviewed the fundamental properties in using BN as a framework for engineering reliability applications and pointed out the present and future research trend. Meanwhile, Uusitalo [117] also

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summarized the advantages and limitations of the BNs application in the field of environmental modelling and management. Extensive introduction to BN can also be found in the textbooks by Pearl [112] and Nielsen and Jensen [118]. Recently, BNs in the context of reliability analysis have also found applications in bridge [119] and marine platform [120], and the results show that BNs have significant advantages over the traditional reliability frameworks. Ma et al. [121] also proposed a BN-based framework for predicting the remaining strength of the entire bridge, by using measurements for individual components, including stiffness, corrosion damage, and load-deflection response. Franchin et al. [122] proposed a BN model to predict the seismic fragility curves of reinforced concrete girder bridges, which enables the performance assessment, upgrade/retrofit interventions and diagnosis of bridge damages in a seismic event. Other examples that are relevant to the proposed research is by Straub and his colleague. Straub [103] developed a dynamic Bayesian network (DBN) based reliability analysis framework which enables the efficient and robust reliability updating for a realistic deterioration due to fatigue crack growth. Later on, Luque and Straub [123] extended the approach to allow the system reliability updating considering the fatigue crack.

It should be mentioned that despite the popularity of using BN/DBN for system reliability analysis of civil structures, there are still some limitations for real engineering practice. For example, acquiring all conditional probabilities P(Xi|pa(Xi)) to get the conditional probability table (CPT) is sometimes not feasible

even with domain experts’ experience. Also, it is difficult to build Bayesian network for a large complex

structure/system and the inference of BN/DBN can become computationally complex even if such a complex BN/DBN (probably with a large number variables) is acquired. In addition, for the research in the field of civil engineering as well as many other fields, the data and parameters are often defined in a continuous space. However, for most applications of BN/DBN models, all the variables are usually discrete due to technicalities of the calculation scheme, which makes their applicability subject to great limitations in reliability analysis.

In recognition of the limitations of BN/DBN as discussed previously, the Bayesian network inference is briefly discussed herein. The BN/DBN enables the calculation of posterior distribution of a set of random variables when new observations are available. This task is also known as Bayesian inference. In general,

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the Bayesian inference algorithms are divided into two categories, (a) exact inference algorithms (e.g., [124]), and (b) approximate inference algorithms which includes deterministic methods (e.g., [125,126]) and probabilistic-based sampling methods (e.g., [127–129]). Interested readers can refer to [118,130] for more details for these inference algorithms. The basic operations that are inherent in the exact inference are restriction, combination and elimination, regardless of which algorithm is implemented during the inference process [113]. One of the most popular algorithms for exact inference is proposed by Lauritzen and Spiegelhalter [124], which consists of two steps, i.e., the moralization of network structure and the triangulation of the resultant moral graph. Nevertheless, the exact inference is sometimes not feasible when dealing with hybrid BN/DBN containing both discrete and continuous variables, and approximate inference is often proposed for inference with hybrid BN/DBN. The most common approach for approximate inference is discretization, i.e., to replace the random variables that are defined in continuous space with equivalent variables that are defined in discrete space. By applying the discretization, the original continuous domain of a random variable divided into discrete intervals and the probability of each interval is calculated based on the conditional or the marginal PDF of the random variable. The choice of discretization intervals is critical in order to achieve high accuracy in the approximations without losing computational performance too much. Several algorithms have been proposed to obtain the optimal intervals in the context of reliability analysis where rare events are of interest [131–133]. Some commonly used probabilistic-based sampling methods include Markov Chain Monte Carlo (MCMC) [127], Gibbs sampling [128], Metropolis-Hastings sampling [129] and importance sampling [134]. Langseth et al. [113] applied and compared four different inference algorithms in a hybrid BN: discretization, mixtures of truncated exponentials (MTEs), variational methods, and MCMC. Based on the results, they concluded that discretization method (with moderate number of regions) is the fastest technique, while MCMC is comparably much slower than all the other three algorithms in order to obtain results of comparable quality. This is especially true that the computational efficiency of MCMC decreases significantly when the number of observations in the BN increases and/or the probability of failure (i.e., rare event in reliability analysis) of interest decreases.

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During the past few decades, a number of BN tools have been developed for computing with BN/DBN such as, Hugin (http://www.hugin.com/), BayesiaLab (http://www.bayesia.com/), Netica (http://www.norsys.com/), BUGS (http://www.mrc-bsu.cam.ac.uk/bugs/), SamIam (http://reasoning.cs.ucla.edu/samiam/), etc. Most of the tools provide BN/DBN inference algorithms such as variable elimination [135,136], junction tree algorithm [118] and Gibbs sampling inference [137], etc.